What is/are your favorite number theory book(s)?
Comments
[deleted] • 10 points • 2024-06-19
What level are you looking for? I really liked the GTM book “A Classical Introduction to Number Theory” by Kenneth Ireland and Rosen. Marcus’s “Number Fields” is also a good book in my opinion. If you want a more advanced treatment, go for “A Course in Arithmetic” by Serre.
[deleted] • 3 points • 2024-06-19
David Burton “Introduction to Elementary Number Theory”, Cohen’s Algebraic Number Theory book, Childress and Milne “Class Field Theory”, Ireland and Rosen “A Classical Introduction to Number Theory”.
attnnah_whisky • 3 points • 2024-06-19
Diamond-Shurman “A First Course in Modular Forms” and Milne “Class Field Theory”
Ill-Room-4895 • 3 points • 2024-06-19
The Number Theory favorites on my bookshelf:
- Elementary Number Theory by David Burton. Clear exposition. Collection of exercises. A pleasure to read. Recommended for all. 448 pages. Rating 4.7 out of 5 on Amazon.
- Number Theory Step by Step by Kuldeep Singh. Many examples. For undergraduates. 400 pages. Rating 4.8 out of 5 on Amazon.
- An Introduction to the Theory of Numbers by Hardy and Wright. The primary and classic text in elementary number theory. Clear exposition, suitable for undergraduates. 644 pages. Rating 4.5 out of 5 on Amazon.
- Introduction to Analytic Number Theory nu Tom Apostal. For undergraduates. Intermediate to advanced readers. 352 pages. Rating 4.7 out of 5 on Amazon.
- Number Theory by Andrej Dujella. For undergraduates, graduates, and researchers in number theory, algebra, and cryptography. 636 pages. Rating 4.8 out of 5 on Amazon.
Mickanos • 1 points • 2024-06-19
Basic Number Theory by Weil approaches number theory from a different angle than most references. It is absolutely not recommended for a first contact with the topic, but pretty enlightening as a second or third reading.
Anything by Serre is usually a cool read.
Voight’s Quaternion Algebras is a modern classic, but too large to read from cover to cover.
miaaasurrounder • 1 points • 2024-08-02
may i ask why is it not considered suitable for begginers?(the first book you’ve mentioned)
Mickanos • 1 points • 2024-08-03
Of course! The thing is, the title of the book obviously seems to contradict what I say, and knowing a bit about Weil’s character, I’m pretty sure it’s an intentional tongue in cheek choice.
The point is, the topic of the book is indeed “basic number theory”, as the book deals with the fundamental concepts of algebraic number theory and class field theory. What is absolutely not basic about it is the framing. Instead of looking for what would be a natural pedagogical beginning for a student discovering algebraic number theory, Weil is going more for the natural mathematical construction. This might work as a first book on number theory for a seasoned mathematician coming from another field, and maybe even for some students with a more bourbakist approach to things. After all, a friend of mine learned commutative algebra reading Bourbaki and learned Algebraic Geometry reading EGA.
But let me put it like this: you are an upper year undergrad student or a new graduate student interested in algebraic number theory. You are familiar with Galois theory, and some commutative algebra. You want to learn about number fields, and what they say about things like polynomial equations over the integers or the rationals, or the behavior of prime numbers. You pick up this book. The first two chapter tell you about locally compact fields and their Haar measure. Then, in the third chapter you finally hear about number fields, but now you’re learning about their embeddings in locally compact fields and their tensor products. In chapter 4, you look at infinite products of locally compact fields, their characters and topological properties. Only in chapter 5 do you look at rings of algebraic integers (but you just call them orders) and their prime ideals. This book was fascinating to discover, but I already knew what a local field and an Adèle ring were before I picked it up.
If you are already familiar with these objects or are ready to stomach unmotivated abstraction, this is pretty cool, but if you just wanted to learn about Pell’s equation, you might find yourself a bit at a loss. In general an introduction to algebraic number will provide you with examples like computing Pythgoeran triples by factoring in Q(i), and later construct the abstract things from Weil’s book on top of a decent understanding of the notions you initially want to care about, such as prime numbers and algebraic integers.
mathemorpheus • 1 points • 2024-06-20
Algebraic Number Theory by Cassels and Froelich
Basic Number Theory by Weil
Local Fields by Serre
Neukirch’s book